Theoretical analysis of regret matching that improves the \(O(\frac{1}{\sqrt{T}})\)  bound has proved difficult, despite a wealth of empirical evidence suggesting that the bound is loose in practice. Nevertheless, there is reason to believe that this correlation is no accident. Fully cooperative games are known to always admit a pure Nash equilibrium [1]}, and thus, regret matching with greedy weights will converge asymptotically after only a constant number of iterations, since greedy weights will give infinite weight to the pure equilibrium the moment it is discovered and the size of the game is constant with respect to the number of iterations run. On the other hand, zero-sum games are more similar to the adversarial case where Theorem REF  proves that convergence can occur no faster than \(O(\frac{1}{\sqrt{T}})\)  There are some subtle differences in that the adversary in Theorem REF  is more powerful than an opponent in a two-player zero-sum game in that she is able to pick any regret vector for each timestep without the constraints of picking from the game matrix..
